5.1. Periodic Signals: A signal f(t) is periodic iff for some T 0 > 0,

Transcription

1 5. Periodic Sigals: A sigal f(t) is periodic iff for some >, f () t = f ( t + ) i t he smallest value that satisfies the above coditios is called the period of f(t). Cosider a sigal examied over to 5 secods show below. What values of satisfy the coditio i the defiitio of periodic? What is the period of this sigal? 2 si(2*pi*t) + si(2*pi*2*t +.3).5 Amplitude Secods

2 5.2 Properties of Periodic Sigals: he itegratio over oe full period has the same result, idepedet of where the itegratio limits begi or ed. a+ b+ g() t dt = g() t dt = g() t dt a b he Siusoid: gt ( ) = si( t + ) = si( ft + ) = 2π ω θ 2π θ si t + θ Phase - θ, Frequecy (radias per secod) - ω, Frequecy (Hz) - f, Period - Harmoics: Cosider a sigal cosistig of a sum of siusoids whose frequecies are iteger multiples of a fudametal frequecy f : g( t) = C + C si( 2πf t + θ ) + C si( 2π 2 f t + θ ) + C si( 2π 3 f t + θ ) + L where f is the fudametal frequecy ad the siusoid with frequecy kf is called the k-th harmoic i the series ad C is the zero-th harmoic (or DC). Examples: Fid the periods ad fudametal frequecies of the sigals below.

3 5.3 a) g(t) = 6si(2πt) -3cos(3πt) (Hit: For f fid the greatest commo factor betwee the frequecies, the least commo factor will also work) (Hit: For fid the least commo multiple betwee the periods) b) g(t) = 4 + 3si(2πt) + si(5πt) + 2si(24πt) c) g(t) = 4cos(8πt) - cos(2πt) + cos(πt) d) g(t) = 6si(3.2πt) -3cos(7.2πt) e) g(t) = 6si(2πt) - 3si(2t) (Hit: Must the sum of periodic siusoidal sigals be periodic?) Fourier Series:

4 5.4 A periodic sigal, f(t), of almost ay form ca be expressed as a series of harmoic siusoidals: rig-forms f ( t) = C + C cos( ω t + θ ) = a + a cos( ω t) + b si( ω t) = = where C = a, a = C cos( θ ), b = C si( θ ), b C a b = 2 + 2, θ = ta a Expoetial-forms f ( t) = D exp( j t ) = ω where D belogs to the set of complex umbers ad D = D * -

5 5.5 Example: Show by plottig the harmoic compoets of a Fourier series that the periodic extesio of r(t+)-2r(t) o the iterval [-, ], has Fourier coefficiets: 2 ( cos( π )) a =, a = b 2, = 2 π ( ) >>t = [-4:.:4]; % Set up time axis to show 4 periods >>s =.5*oes(size(t)); % Iitialize summig array with a >>for k=:, % Loop to sum up harmoics >>s = s + 2*((-cos(k*pi))/(pi*k)^2)*cos(k*pi*t); >>ed >>plot(t,s,'w') first harmoic Up to -th harmoic.5 first 5 harmoics time time time Up to third harmoic Up to 5-st harmoic time time

6 5.6 Computig Fourier Series Coefficiets (trig-form): Give the Fourier Series represetatio of a periodic sigal f(t): f () t = a + a cos( ω t) + b si( ω t) = it ca be show that the coefficiets for f(t) ca be computed with: 2 2 a = f t dt a f t t dt b f t t dt (), = ()cos( ω ), = ()si( ω ) Derive the Fourier Series coefficiets i trig form for the waveform i the previous example. Derive the Fourier Series coefficiets i trig form for the periodic extesio of o the iterval [, ], where <Δ< p t Δ () show that: a si( ω Δ) ( cos( ω π )) =, a = b =, ( π Δ) ( π Δ)

7 5.7 Computig Fourier Series Coefficiets (expoetial-form): Give the Fourier Series represetatio of a periodic sigal f(t): f ( t) = D exp( j t ) = it ca be show that the coefficiets for f(t) ca be computed with: D = f t j t dt ( )exp( ω ) ω Note the differeces betwee the rig ad Expoetial form:. Coefficiets for the trig for are real for real f(t), while for the expoetial form they are complex. 2. he idex for summatio is from to for the trig form ad from - to + for the expoetial form 3. he formulas for the expoetial form are more compact ad easier to work with tha those for the trig form. he coefficiets for compact trig represetatio ca be determie from the expoetial form coefficiets: C = D, ad C = 2 D, θ = D for > Examples:

8 5.8 Derive the Fourier Series coefficiets i expoetial form for the periodic extesio of p () t o the iterval [, Δ ], where <Δ<. Plot the magitudes ad phases of the coefficiets (for = ms ad Δ = ms) as a fuctio of frequecy (f) ad discuss their meaig. j Δπ πδ exp si Show: D = for D = π Δ >>t = e-3 % Period >>del = e-3 % Duratio of pulse >> = [-6:-, eps, :6]; % Harmoic idex (eps is close to zero) >>f = /t; % Fudametal Frequecy >>d = exp(-j*pi*del*f*).*si(f**pi*del)./(pi*del*); % Fourier Coefficiets >>figure() >>stem(f*,abs(d),'-w') % Stem plot of magitudes >>title('magitude of Fourier Coefficiets') >>xlabel('hz') >>figure(2) >>stem(f*,agle(d),'-w') % Stem plot of phases >>title('phase of Fourier Coefficiets') >>xlabel('hz')

9 5.9 M agitude of Fourier C oefficiets Hz 4 P hase of F ourier C oefficiets Hz

10 5. Examples: Derive the Fourier Series coefficiets i expoetial form for the periodic extesio of r(t) o the iterval [,.5]. Plot the magitudes ad phases of the coefficiets as a fuctio of frequecy (f) ad discuss their meaig. j Show: D = 4π for D = 4 >>t =.5 % Period >> = [-3:-, eps, :3]; % Harmoic idex (eps is close to zero) >>f = /t; % Fudametal Frequecy >>d = j./(4*pi*); % Fourier Coefficiets. >>d(3) = /4; % Must substitute DC term i directly, sice expressio was >> % simplified beyod the poit of usig L'Hopital's rule >>figure() >>stem(f*,abs(d),'-w') % Stem plot of magitudes >>title('magitude of Fourier Coefficiets') >>xlabel('hz') >>figure(2) >>stem(f*,agle(d),'-w') % Stem plot of phases >>title('phase of Fourier Coefficiets') >>xlabel('hz')

11 5..25 M a g itud e o f F o urie r C o e fficie ts Hz Phase of Fourier Coefficiets Hz